
\({\mathcal {M}}(k)\) :
\({\mathcal {M}}(k):=\bigcup _{\alpha \in {\mathbb {N}}}\bigcup _{\beta \in {\mathbb {N}}}{\mathcal {M}}_{k^{\alpha }\times k^{\beta }},\) 
where \(k\in {\mathbb {N}}\)  and \(k>1\) .
It is obvious that \({\mathcal {M}}(k)~ < ~{\mathcal {M}}\) . (In this section \(A<B\)  means \(A\)  is a submonoid of \(B\) ). This sub-monoid is useful for calculating the product of tensors over \(k\)  dimensional vector space [1]}. It is particularly useful for \(k\) -valued logical dynamic systems [2]}, [3]}. When \(k=2\) , it is used for Boolean dynamic systems.
In this sub-monoid the STP can be defined as follows:

